FCLoopFindIntegralMappings[{int1, int2, ...}, {p1, p2, ...}]
finds mappings between scalar multiloop integrals
int1, int2, ... that depend on the loop momenta
p1, p2, ... using the algorithm of Alexey Pak arXiv:1111.0868.
The current implementation is based on the
FindEquivalents function from FIRE 6 arXiv:1901.07808
It is also possible to invoke the function as
FCLoopFindIntegralMappings[{GLI[...], ...}, {FCTopology[...], ...}] or FCLoopFindIntegralMappings[{FCTopology[...], ...}].
Notice that in this case the value of the option
FinalSubstitutions is ignored, as replacement rules will be
extracted directly from the definition of the topology.
The default output is a list of two lists. The first list contains mapping rules between different loop integrals, while the second list provides all unique master integrals extracted from the input.
An alternative output mode is activated when the option
List is set to True. In this case the output
is a list of lists, where each list contains master integrals that were
identified to be identical.
The option PreferredIntegrals can be used to enforce the
mapping onto a preferred set of master integral. Notice that the final
result will only contain those preferred integrals, that are actually
present in the input.
Overview, FCTopology, GLI, FCLoopToPakForm, FCLoopPakOrder, FCLoopFindTopologyMappings
When given a list of FeynAmpDenominator-integrals, the
function will merely group identical integrals into sublists
ints = {FAD[{p1, m1}], FAD[{p1 + q, m1}], FAD[{p1, m2}]}\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{(\text{p1}+q)^2-\text{m1}^2},\frac{1}{\text{p1}^2-\text{m2}^2}\right\}
FCLoopFindIntegralMappings[ints, {p1}]\left\{\left\{\frac{1}{\text{p1}^2-\text{m1}^2},\frac{1}{(\text{p1}+q)^2-\text{m1}^2}\right\},\left\{\frac{1}{\text{p1}^2-\text{m2}^2}\right\}\right\}
The following 3 integrals look rather different from each other, but are actually identical
ints = {FAD[p1] FAD[p1 - p3 - p4] FAD[p4] FAD[p3 + q1] FAD[{p3, m1}] FAD[{p1 - p4, m1}]*
FAD[{p1 + q1, 0}, {p1 + q1, 0}], FAD[p4] FAD[p1 - p3 + q1] FAD[p3 + q1] FAD[p1 + p4 + q1]*
FAD[{p3, m1}] FAD[{p1 + q1, m1}] FAD[{p1 + p4 + 2 q1, 0}, {p1 + p4 + 2 q1, 0}],
FAD[p1] FAD[p4 - 2 q1] FAD[p3 + q1] FAD[p1 - p3 - p4 + 2 q1] FAD[{p3, m1}]*
FAD[{p1 - p4 + 2 q1, m1}] FAD[{p1 + q1, 0}, {p1 + q1, 0}]}\left\{\frac{1}{\text{p1}^2 \;\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}-\text{p4})^2 \left((\text{p1}-\text{p4})^2-\text{m1}^2\right)},\frac{1}{\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}+\text{q1})^2 (\text{p1}+\text{p4}+\text{q1})^2 (\text{p1}+\text{p4}+2 \;\text{q1})^4 \left((\text{p1}+\text{q1})^2-\text{m1}^2\right)},\frac{1}{\text{p1}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p4}-2 \;\text{q1})^2 (\text{p1}-\text{p3}-\text{p4}+2 \;\text{q1})^2 \left((\text{p1}-\text{p4}+2 \;\text{q1})^2-\text{m1}^2\right)}\right\}
FCLoopFindIntegralMappings[ints, {p1, p3, p4}]\left( \begin{array}{ccc} \frac{1}{\text{p1}^2 \;\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}-\text{p4})^2 \left((\text{p1}-\text{p4})^2-\text{m1}^2\right)} & \frac{1}{\text{p4}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p3}+\text{q1})^2 (\text{p1}-\text{p3}+\text{q1})^2 (\text{p1}+\text{p4}+\text{q1})^2 (\text{p1}+\text{p4}+2 \;\text{q1})^4 \left((\text{p1}+\text{q1})^2-\text{m1}^2\right)} & \frac{1}{\text{p1}^2 \left(\text{p3}^2-\text{m1}^2\right) (\text{p1}+\text{q1})^4 (\text{p3}+\text{q1})^2 (\text{p4}-2 \;\text{q1})^2 (\text{p1}-\text{p3}-\text{p4}+2 \;\text{q1})^2 \left((\text{p1}-\text{p4}+2 \;\text{q1})^2-\text{m1}^2\right)} \\ \end{array} \right)
If the input is a list of GLI-integrals,
FCLoopFindIntegralMappings will return a list containing
two sublists. The former will be a list of replacement rules while the
latter will contain all unique master integrals
ClearAll[topo1, topo2];
topos = {
FCTopology[topo1, {SFAD[{p1, m^2}], SFAD[{p2, m^2}]}, {p1, p2}, {}, {}, {}],
FCTopology[topo2, {SFAD[{p3, m^2}], SFAD[{p4, m^2}]}, {p3, p4}, {}, {}, {}]
}\left\{\text{FCTopology}\left(\text{topo1},\left\{\frac{1}{(\text{p1}^2-m^2+i \eta )},\frac{1}{(\text{p2}^2-m^2+i \eta )}\right\},\{\text{p1},\text{p2}\},\{\},\{\},\{\}\right),\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{(\text{p3}^2-m^2+i \eta )},\frac{1}{(\text{p4}^2-m^2+i \eta )}\right\},\{\text{p3},\text{p4}\},\{\},\{\},\{\}\right)\right\}
glis = {GLI[topo1, {1, 1}], GLI[topo1, {1, 2}], GLI[topo1, {2, 1}], GLI[topo2, {1, 1}],
GLI[topo2, {2, 2}]}\left\{G^{\text{topo1}}(1,1),G^{\text{topo1}}(1,2),G^{\text{topo1}}(2,1),G^{\text{topo2}}(1,1),G^{\text{topo2}}(2,2)\right\}
FCLoopFindIntegralMappings[glis, topos]\left\{\left\{G^{\text{topo2}}(1,1)\to G^{\text{topo1}}(1,1),G^{\text{topo1}}(2,1)\to G^{\text{topo1}}(1,2)\right\},\left\{G^{\text{topo1}}(1,1),G^{\text{topo1}}(1,2),G^{\text{topo2}}(2,2)\right\}\right\}
This behavior can be turned off by setting the value of the option
List to True
FCLoopFindIntegralMappings[glis, topos, List -> True]\left\{\left\{G^{\text{topo1}}(1,1),G^{\text{topo2}}(1,1)\right\},\left\{G^{\text{topo1}}(1,2),G^{\text{topo1}}(2,1)\right\},\left\{G^{\text{topo2}}(2,2)\right\}\right\}
In practice, one usually has a list of preferred integrals onto which
one would like to map the occurring master integrals. Such integrals can
be specified via the PreferredIntegrals option
FCLoopFindIntegralMappings[glis, topos, PreferredIntegrals -> {GLI[topo2, {1, 1}],
GLI[topo2, {2, 1}]}]\left( \begin{array}{ccc} G^{\text{topo1}}(1,1)\to G^{\text{topo2}}(1,1) & G^{\text{topo1}}(1,2)\to G^{\text{topo2}}(2,1) & G^{\text{topo1}}(2,1)\to G^{\text{topo2}}(2,1) \\ G^{\text{topo2}}(1,1) & G^{\text{topo2}}(2,1) & G^{\text{topo2}}(2,2) \\ \end{array} \right)
The indices of GLIs do not have to be integers
topos = {
FCTopology[prop2LtopoG20, {SFAD[{{p1, 0}, {0, 1}, 1}],
SFAD[{{p1 + q1, 0}, {m3^2, 1}, 1}], SFAD[{{p3, 0}, {0, 1}, 1}],
SFAD[{{p3 + q1, 0}, {0, 1}, 1}], SFAD[{{p1 - p3, 0}, {0, 1}, 1}]},
{p1, p3}, {q1}, {}, {}],
FCTopology[prop2LtopoG21, {SFAD[{{p1, 0}, {m1^2, 1}, 1}],
SFAD[{{p1 + q1, 0}, {m3^2, 1}, 1}],
SFAD[{{p3, 0}, {0, 1}, 1}], SFAD[{{p3 + q1, 0}, {0, 1}, 1}],
SFAD[{{p1 - p3, 0}, {0, 1}, 1}]}, {p1, p3}, {q1}, {}, {}]
}\left\{\text{FCTopology}\left(\text{prop2LtopoG20},\left\{\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2-\text{m3}^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{((\text{p1}-\text{p3})^2+i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\{\},\{\}\right),\text{FCTopology}\left(\text{prop2LtopoG21},\left\{\frac{1}{(\text{p1}^2-\text{m1}^2+i \eta )},\frac{1}{((\text{p1}+\text{q1})^2-\text{m3}^2+i \eta )},\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{((\text{p3}+\text{q1})^2+i \eta )},\frac{1}{((\text{p1}-\text{p3})^2+i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\{\},\{\}\right)\right\}
FCLoopFindIntegralMappings[{GLI[prop2LtopoG21, {0, n1, n2, n3, n4}],
GLI[prop2LtopoG20, {0, n1, n2, n3, n4}]}, topos]\left( \begin{array}{c} G^{\text{prop2LtopoG21}}(0,\text{n1},\text{n2},\text{n3},\text{n4})\to G^{\text{prop2LtopoG20}}(0,\text{n1},\text{n2},\text{n3},\text{n4}) \\ G^{\text{prop2LtopoG20}}(0,\text{n1},\text{n2},\text{n3},\text{n4}) \\ \end{array} \right)
It is also possible to find mappings for factorizing integrals, provided that suitable products of integrals are given as preferred integrals
topos = {FCTopology[prop2Ltopo31313, {SFAD[{{
I p1, 0}, {-m3^2, -1}, 1}], SFAD[{{I (p1 + q1), 0}, {-m1^2, -1}, 1}], SFAD[{{
I p3, 0}, {-m3^2, -1}, 1}], SFAD[{{I
(p3 + q1), 0}, {-m1^2, -1}, 1}], SFAD[{{
I (p1 - p3), 0}, {-m3^2, -1}, 1}]}, {p1, p3}, {q1}, {SPD[q1, q1] -> m1^2}, {}],
FCTopology[tad1Ltopo2, {SFAD[{{I p1, 0}, {-m3^2, -1}, 1}]}, {p1}, {}, {SPD[q1,q1] -> m1^2}, {}]}\left\{\text{FCTopology}\left(\text{prop2Ltopo31313},\left\{\frac{1}{(-\text{p1}^2+\text{m3}^2-i \eta )},\frac{1}{(-(\text{p1}+\text{q1})^2+\text{m1}^2-i \eta )},\frac{1}{(-\text{p3}^2+\text{m3}^2-i \eta )},\frac{1}{(-(\text{p3}+\text{q1})^2+\text{m1}^2-i \eta )},\frac{1}{(-(\text{p1}-\text{p3})^2+\text{m3}^2-i \eta )}\right\},\{\text{p1},\text{p3}\},\{\text{q1}\},\left\{\text{q1}^2\to \;\text{m1}^2\right\},\{\}\right),\text{FCTopology}\left(\text{tad1Ltopo2},\left\{\frac{1}{(-\text{p1}^2+\text{m3}^2-i \eta )}\right\},\{\text{p1}\},\{\},\left\{\text{q1}^2\to \;\text{m1}^2\right\},\{\}\right)\right\}
Here we ask the function to map all products of two 1-loop tadpoles
to GLI[tad1Ltopo2,{1}]^2
FCLoopFindIntegralMappings[{GLI[tad1Ltopo2, {1}]^2,
GLI[prop2Ltopo31313, {0, 0, 1, 0, 1}]}, topos, PreferredIntegrals -> {GLI[tad1Ltopo2, {1}]^2}]\left( \begin{array}{c} G^{\text{prop2Ltopo31313}}(0,0,1,0,1)\to G^{\text{tad1Ltopo2}}(1)^2 \\ G^{\text{tad1Ltopo2}}(1)^2 \\ \end{array} \right)